Ensemble Statistics Research Articles

Clustering-Based Spatial Interpolation of Parametric Postprocessing Models

Abstract Since the start of the operational use of ensemble prediction systems, ensemble-based probabilistic forecasting has become the most advanced approach in weather prediction. However, despite the persistent development of the last three decades, ensemble forecasts still often suffer from the lack of calibration and might exhibit systematic bias, which calls for some form of statistical postprocessing. Nowadays, one can choose from a large variety of postprocessing approaches, where parametric methods provide full predictive distributions of the investigated weather quantity. Parameter estimation in these models is based on training data consisting of past forecast-observation pairs; thus, postprocessed forecasts are usually available only at those locations where training data are accessible. We propose a general clustering-based interpolation technique of extending calibrated predictive distributions from observation stations to any location in the ensemble domain where there are ensemble forecasts at hand. Focusing on the ensemble model output statistics (EMOS) postprocessing technique, in a case study based on 10-m wind speed ensemble forecasts of the European Centre for Medium-Range Weather Forecasts, we demonstrate the predictive performance of various versions of the suggested method and show its superiority over the regionally estimated and interpolated EMOS models and the raw ensemble forecasts as well.

Effect of attention on ensemble perception: Comparison between exogenous attention, endogenous attention, and depth.

Ensemble perception is an important ability of human beings that allows one to extract summary information for scenes and environments that contain information that far exceeds the processing limit of the visual system. Although attention has been shown to bias ensemble perception, two important questions remain unclear: (1) whether direct manipulations on different types of spatial attention could produce similar effects on ensembles and (2) whether factors potentially influencing the attention distribution, such as depth perception, could evoke an indirect effect of attention on ensemble representation. This study aims to address these questions. In Experiments 1 and 2, two types of precues were used to evoke exogenous and endogenous attention, respectively, and the ensemble color perceptions were examined. We found that both exogenous and endogenous attention biased ensemble representation towards the attended items, and the latter produced a greater effect. In Experiments 3 and 4, we examined whether depth perception could affect color ensembles by indirectly influencing attention allocation in 3D space. The items were separated in two depth planes, and no explicit cues were applied. The results showed that color ensemble was biased to closer items when depth information was task relevant. This suggests that ensemble perception is naturally biased in 3D space, probably through the mechanism of attention. Computational modeling consistently showed that attention exerted a direct shift on the ensemble statistics rather than averaging the feature values over the cued and noncued items, providing evidence against an averaging process of individual perception.

Towards a 2DEnVar surface data assimilation approach within the convective scale numerical weather prediction model AROME‐France

AbstractSurface data assimilation (DA) plays a key role in the initialisation of soil variables in coupled surface–atmosphere numerical weather prediction (NWP) models. Météo‐France, as many other NWP centres, uses an operational surface DA method based on an optimal interpolation (OI), implemented more than 30 years ago. The current surface DA system implemented at Météo‐France in the operational limited‐area AROME‐France model operates in two steps. First, a two‐dimensional OI method analyses the diagnostic screen‐level parameters, which are then used to initialise the prognostic soil variables according to a one‐dimensional OI method. This article presents the implementation and the results of a new two‐dimensional ensemble‐based variational (2DEnVar) system, using the object‐oriented prediction system framework, for the diagnostic screen‐level variable analyses (2‐m temperature and 2‐m relative humidity) within the AROME‐France model. These analysed variables are then used to initialise the soil temperature and soil moisture of the first and second soil layers. A two‐dimensional variational system is initially implemented and compared with an equivalent system that uses an OI surface analysis. The comparison of 2‐m temperature increments shows that the two systems behave in a similar way. From this two‐dimensional variational approach, a 2DEnVar scheme is developed and compared with the OI screen‐level analysis used operationally (reference system). The results are encouraging even though a degradation in forecast quality is noticed at short ranges for 2‐m temperature and 2‐m relative humidity. This is mainly due to a low spread near the surface in the ensemble‐based data assimilation that is used to compute background‐error covariances in 2DEnVar. Sensitivity tests have been performed to improve the initial version of the 2DEnVar scheme. A positive and significant impact is noticed when a multiplicative inflation is applied to the background perturbations coming from the ensemble‐based data assimilation for the computation of 2‐m temperature and 2‐m relative humidity background‐error covariances. Finally, when comparing the inflated 2DEnVar system with the reference, the two systems behave similarly. This suggests that the 2DEnVar scheme, based on ensemble statistics, is capable of properly reconstructing background‐error structures derived from the empirical correlations described in OI.

Spectral fluctuations of multiparametric complex matrix ensembles: evidence of a single parameter dependence

Abstract We numerically analyze the spectral statistics of the multiparametric Gaussian ensembles of complex matrices with zero mean and variances with different decay routes away from the diagonals. As the latter mimics different degree of effective sparsity among the matrix elements, such ensembles can serve as good models for a wide range of phase transitions e.g. localization to delocalization in non-Hermitian systems or Hermitian to non-Hermitian one. Our analysis reveals a rich behavior hidden beneath the spectral statistics e.g. a crossover of the spectral statistics from Poisson to Ginibre universality class with changing variances for finite matrix size, an abrupt transition for infinite matrix size and the role of complexity parameter, a single functional of all system parameters, as a criteria to determine critical point. We also confirm the theoretical predictions in \cite{psgs, psnh}, regarding the universality of the spectral statistics in non-equilibrium regime of non-Hermitian systems characterized by the complexity parameter.

A method for evaluating deep generative models of images for hallucinations in high-order spatial context

Deep generative models (DGMs) have the potential to revolutionize diagnostic imaging. Generative adversarial networks (GANs) are one kind of DGM which are widely employed. The overarching problem with deploying any sort of DGM in mission-critical applications is a lack of adequate and/or automatic means of assessing the domain-specific quality of generated images. In this work, we demonstrate several objective and human-interpretable tests of images output by two popular DGMs. These tests serve two goals: (i) ruling out DGMs for downstream, domain-specific applications, and (ii) quantifying hallucinations in the expected spatial context in DGM-generated images. The designed datasets are made public and the proposed tests could also serve as benchmarks and aid the prototyping of emerging DGMs. Although these tests are demonstrated on GANs, they can be employed as a benchmark for evaluating any DGM. Specifically, we designed several stochastic context models (SCMs) of distinct image features that can be recovered after generation by a trained DGM. Together, these SCMs encode features as per-image constraints in prevalence, position, intensity, and/or texture. Several of these features are high-order, algorithmic pixel-arrangement rules which are not readily expressed in covariance matrices. We designed and validated statistical classifiers to detect specific effects of the known arrangement rules. We then tested the rates at which two different DGMs correctly reproduced the feature context under a variety of training scenarios, and degrees of feature-class similarity. We found that ensembles of generated images can appear largely accurate visually, and show high accuracy in ensemble measures, while not exhibiting the known spatial arrangements. The main conclusion is that SCMs can be engineered, and serve as benchmarks, to quantify numerous per image errors, i.e., hallucinations, that may not be captured in ensemble statistics but plausibly can affect subsequent use of the DGM-generated images.

Time‐series‐based ensemble model output statistics for temperature forecasts postprocessing

AbstractThe uncertainty in numerical weather prediction models is nowadays quantified by the use of ensemble forecasts. Although these forecasts are continuously improved, they still suffer from systematic bias and dispersion errors. Statistical postprocessing methods, such as the ensemble model output statistics (EMOS), have been shown to substantially correct the forecasts. This work proposes an extension of EMOS in a time‐series framework. Besides taking account of seasonality and trend in the location and scale parameter of the predictive distribution, the autoregressive process in the mean forecast errors or the standardized forecast errors is considered. The models can be further extended by allowing generalized autoregressive conditional heteroscedasticity. Furthermore, it is outlined how to use these models for arbitrary forecast horizons. To illustrate the performance of the suggested EMOS models in time‐series fashion, we present a case study for the postprocessing of 2 m surface temperature forecasts using five different lead times and a set of observation stations in Germany. The results indicate that the time‐series EMOS extensions are able to significantly outperform the benchmark models EMOS and autoregressive EMOS (AR‐EMOS) in most of the lead time–station cases.

Incorporating memory effect into a fractional stochastic diffusion particle tracking model for suspended sediment using Malliavin-Calculus-based fractional Brownian Motion

This study aims to develop a Fractional Stochastic Diffusion Particle Tracking Model (FSDPTM) to illustrate suspended sediment’s long-term dependence (LTD) trajectories immersed in fully developed turbulent flow. Carried by eddies in the open channel, individual suspended sediment behavior exhibits dependence on its past increments. The Fractional Brownian Motion (FBM) is incorporated into the FSDPTM to account for the randomness of turbulent flow and to include the time-dependent increments as a memory effect. Additionally, to enhance the capacity for differentiating FBM, this study introduces Malliavin Calculus, the stochastic calculus of variation, to construct the square -differentiable FBM. For the numerical realizations of square differentiable FBM, the integration of fractional Gaussian white noise and Wiener–Ito Chaos expansion with the Hermite sequence is employed. The FSDPTM offers probabilistic LTD trajectories as solutions to the Ito-type Langevin equation. It also provides particle velocity, acceleration, and ensemble statistics, collectively demonstrating the application of FBM to sediment transport modeling. In conclusion, the key feature of the FSDPTM is its ability to translate LTD derived from FBM into the trajectory of suspended sediment particles in fully developed turbulent flow. Additionally, it expands the solutions from non-differentiable to square-differentiable stochastic processes.

Ensemble Variability Properties of Active Galactic Nuclei in the SDSS DR17

We present the results from a study of ∼9600 broadline-selected active galactic nuclei (AGNs) with host galaxies detected from the Sloan Digital Sky Survey (SDSS) Data Release 17. We compute ensemble variability statistics based on a comparison of the original SDSS photometric data with spectrophotometric measurements obtained days to decades later in the Sloan g, r, and i bands. Galaxy and AGN templates have been fitted to the SDSS spectra to isolate the AGN component from the host galaxy. The sources have absolute magnitudes in the range −24 < M i < −18 and lie at redshifts less than z ∼ 0.9. A variability analysis reveals that the anticorrelation between luminosity and variability amplitude continues down to log(Lbol [erg s−1]) = 43.5, demonstrating that the relationship extends by 4 orders of magnitude in AGN luminosity. To further explore the connection between AGN luminosity and variability, we determine the black hole mass and the accretion rate through measurement of the Hβ line width and the monochromatic luminosity at rest frame 5100 Å. Our results suggest that the accretion rate is the dominant parameter impacting the amplitude of variability and that the anticorrelation between the accretion rate and amplitude extends to rates as low as 1% Eddington. Moreover, we also identify an anticorrelation between the variability amplitude and black hole mass, with the correlation appearing strongest among the AGNs with low accretion rates.

Significant advancement in subseasonal-to-seasonal summer precipitation ensemble forecast skills in China mainland through an innovative hybrid CSG-UNET method

Reliable Subseasonal-to-Seasonal (S2S) forecasts of precipitation are critical for disaster prevention and mitigation. In this study, an innovative hybrid method CSG-UNET combining the UNET with the censored and shifted gamma distribution based ensemble model output statistic (CSG-EMOS), is proposed to calibrate the ensemble precipitation forecasts from ECMWF over the China mainland during boreal summer. Additional atmospheric variable forecasts and the data augmentation are also included to deal with the potential issues of low signal-to-noise ratio and relatively small sample sizes in traditional S2S precipitation forecast correction. The hybrid CSG-UNET exhibits a notable advantage over both individual UNET and CSG-EMOS in improving ensemble precipitation forecasts, simultaneously improving the forecast skills for lead times of 1–2 weeks and further extending the effective forecast timeliness to ∼4 weeks. Specifically, the climatology-based Brier Skill Scores are improved by ∼0.4 for the extreme precipitation forecasts almost throughout the whole timescale compared with the ECMWF. Feature importance analyze towards CSG-EMOS model indicates that the atmospheric factors make great contributions to the prediction skill with the increasing lead times. The CSG-UNET method is promising in subseasonal precipitation forecasts and could be applied to the routine forecast of other atmospheric and ocean phenomena in the future.

Comparative temporal dynamics of individuation and perceptual averaging using a biological neural network model

Analyzing visual scenes and computing ensemble statistics, known as perceptual averaging, is crucial for the stable sensory experience of a cognitive agent. Despite the apparent simplicity of applying filters to scenes, the challenge arises from our brain’s seamless transition between summarization and individuation across various reference frames (retinotopic, spatiotopic, and hemispheric). In this study, we explore the capability of a neural network to dynamically switch between individuation and summarization. Our chosen computational model, a fully connected on-center off-surround recurrent neural network previously employed for enumeration/individuation, demonstrates the potential to extract both summary statistics and achieve high individuation accuracy. Notably, our results show that the individuation accuracy can reach close to perfection within a presentation duration of 100 ms, but not so for summarization. We have also shown a spatially varying excitation version of the network that can explain quite a few interesting spatio-temporal patterns of perception. These findings not only highlight the feasibility of such a neural network but also provide insights into the temporal dynamics of ensemble perception.

Deterministic Forecasting and Probabilistic Post‐Processing of Short‐Term Wind Speed Using Statistical Methods

AbstractThere is a great need for an accurate short‐term wind speed forecast, and statistical forecasts have gained increased popularity for their computational efficiency and satisfactory skill. However, there has been no systematic research to fully explore the capabilities of statistical approaches and evaluate the applicability of probabilistic information from statistical ensemble. This study first compares the skills of different statistical methods, based on linear regression, machine learning (ML), and deep learning (DL), using three strategies (i.e., direct, recursive, and multi‐output) against the three operational numerical models and their bias‐corrections, for short‐term wind speed forecast over Pearl River Estuary during 2018–2021. Inter‐comparison between statistical forecasts reveals the dominant superiority of direct strategy. On this basis, Random Forest (RF) and Support Vector Machines (SVM) perform best compared to other statistical forecasts and bias correction of numerical forecasts throughout 48 hr lead time, while the performance of methods with simplified (linear) or more complex (DL) model structures degrades significantly. Moreover, the top 10 forecasts are utilized to account for forecast uncertainties but present a substantial under‐dispersed prediction. Two traditional methods and three modern methods are implemented to perform probabilistic post‐processing. Modern methods based on ML or DL present worse skills, while traditional methods, particularly for ensemble model output statistics, show added value in discriminating binary events due to limited enhancements in calibration. Overall, RF and SVM using direct strategy are highly recommended for short‐term wind speed forecasts, and efforts are ongoing to address the issues of strong wind prediction and ensemble calibration.

Accounting for residual errors in atmosphere–ocean background models applied in satellite gravimetry

The Atmosphere and Ocean non-tidal De-aliasing Level-1B (AOD1B) product is widely used in precise orbit determination and satellite gravimetry to correct for transient effects of atmosphere–ocean mass variability that would otherwise alias into monthly mean global gravity fields. The most recent release is based on the global ERA5 reanalysis and ECMWF operational data together with simulations from the general ocean circulation model MPIOM consistently forced with fields from the corresponding atmospheric dataset. As background models are inevitably imperfect, residual errors will consequently propagate into the resulting geodetic products. Accounting for uncertainties of the background model data in a statistical sense, however, has been shown before to be a useful approach to mitigate the impact of residual errors leading to temporal aliasing artefacts. In light of the changes made in the new release RL07 of AOD1B, previous uncertainty assessments are deemed too pessimistic and thus need to be revisited. We here present an analysis of the residual errors in AOD1B RL07 based on ensemble statistics derived from different atmospheric reanalyses, including ERA5, MERRA2 and JRA55. For the oceans, we investigate the impact of both the forced and intrinsic variability through differences in MPIOM simulation experiments. The atmospheric and oceanic information is then combined to produce a new time-series of true errors, called AOe07, which is applicable in combination with AOD1B RL07. AOe07 is further complemented by a new spatial error variance–covariance matrix. Results from gravity field recovery simulation experiments for the planned Mass-Change and Geosciences International Constellation (MAGIC) based on GFZ’s EPOS software demonstrate improvements that can be expected from rigorously implementing the newly available stochastic information from AOD1B RL07 into the gravity field estimation process.

Calibrated EMOS: applications to temperature and wind speed forecasting

Ensembles of meteorological quantities obtained from numerical models can be used for forecasting weather variables. Unfortunately, such ensembles are often biased and under-dispersed and therefore need to be post-processed. Ensemble model output statistics (EMOS) is a widely used post-processing technique to reduce bias and dispersion errors of numerical ensembles. In the EMOS approach, a full probabilistic prediction is given in the form of a predictive distribution with parameters depending on the ensemble forecast members. Parameters are then estimated and substituted, thus obtaining a so-called estimative predictive distribution. Nonetheless, estimative distributions may perform poorly in terms of the coverage probability of the corresponding quantiles. This work proposes the use of predictive distributions based on a bootstrap adjustment of estimative predictive distributions, in the context of EMOS models. These distributions are calibrated, which means that the corresponding quantiles provide exact coverage probabilities, in contrast to the estimative distributions. The introduction of the bootstrap calibrated procedure for EMOS is the innovative aspect of this study. The performance of the suggested calibrated EMOS is evaluated in two simulation studies, comparing the different predictive distributions by means of the log-score, the continuous ranked probability score, and the coverage of the corresponding predictive quantiles. The results of these simulation studies show that the proposed calibrated predictive distributions improve estimative solutions, both reducing the mean scores and producing quantiles with exact coverage levels. The good performance of the new calibrated EMOS is further stressed in two real data applications, one about maximum daily temperatures at sites located in the Veneto region (Italy) and the other one about wind speed forecasts at weather stations over Germany.

Filtering dynamical systems using observations of statistics.

We consider the problem of filtering dynamical systems, possibly stochastic, using observations of statistics. Thus, the computational task is to estimate a time-evolving density ρ(v,t) given noisy observations of the true density ρ†; this contrasts with the standard filtering problem based on observations of the state v. The task is naturally formulated as an infinite-dimensional filtering problem in the space of densities ρ. However, for the purposes of tractability, we seek algorithms in state space; specifically, we introduce a mean-field state-space model, and using interacting particle system approximations to this model, we propose an ensemble method. We refer to the resulting methodology as the ensemble Fokker-Planck filter (EnFPF). Under certain restrictive assumptions, we show that the EnFPF approximates the Kalman-Bucy filter for the Fokker-Planck equation, which is the exact solution to the infinite-dimensional filtering problem. Furthermore, our numerical experiments show that the methodology is useful beyond this restrictive setting. Specifically, the experiments show that the EnFPF is able to correct ensemble statistics, to accelerate convergence to the invariant density for autonomous systems, and to accelerate convergence to time-dependent invariant densities for non-autonomous systems. We discuss possible applications of the EnFPF to climate ensembles and to turbulence modeling.

Average spectral density of multiparametric Gaussian ensembles of complex matrices

A statistical description of part of a many body system often requires a non-Hermitian random matrix ensemble with nature and strength of randomness sensitive to underlying system conditions. For the ensemble to be a good description of the system, the ensemble parameters must be determined from the system parameters. This in turn makes its necessary to analyze a wide range of multi-parametric ensembles with different kinds of matrix elements distributions. The spectral statistics of such ensembles is not only system-dependent but also non-ergodic as well as non-‘stationary’. A change in system conditions can cause a change in the ensemble parameters resulting an evolution of the ensemble density and it is not sufficient to know the statistics for a given set of system conditions. This motivates us to theoretically analyze a multiparametric evolution of the ensemble averaged spectral density of a multiparametric Gaussian ensemble on the complex plane. Our analysis reveals the existence of an evolutionary route common to the ensembles belonging to same global constraint class and thereby derives a complexity parameter dependent formulation of the spectral density for the non-equilibrium regime of the spectral statistics, away from Ginibre equilibrium limit.

Ensemble analysis and forecast of ecosystem indicators in the North Atlantic using ocean colour observations and prior statistics from a stochastic NEMO–PISCES simulator

Abstract. This study is anchored in the H2020 SEAMLESS project (https://www.seamlessproject.org, last access: 29 January 2024), which aims to develop ensemble assimilation methods to be implemented in Copernicus Marine Service monitoring and forecasting systems, in order to operationally estimate a set of targeted ecosystem indicators in various regions, including uncertainty estimates. In this paper, a simplified approach is introduced to perform a 4D (space–time) ensemble analysis describing the evolution of the ocean ecosystem. An example application is provided, which covers a limited time period in a limited subregion of the North Atlantic (between 31 and 21∘ W, between 44 and 50.5∘ N, between 15 March and 15 June 2019, at a 1/4∘ and a 1 d resolution). The ensemble analysis is based on prior ensemble statistics from a stochastic NEMO (Nucleus for European Modelling of the Ocean)–PISCES simulator. Ocean colour observations are used as constraints to condition the 4D prior probability distribution. As compared to classic data assimilation, the simplification comes from the decoupling between the forward simulation using the complex modelling system and the update of the 4D ensemble to account for the observation constraint. The shortcomings and possible advantages of this approach for biogeochemical applications are discussed in the paper. The results show that it is possible to produce a multivariate ensemble analysis continuous in time and consistent with the observations. Furthermore, we study how the method can be used to extrapolate analyses calculated from past observations into the future. The resulting 4D ensemble statistical forecast is shown to contain valuable information about the evolution of the ecosystem for a few days after the last observation. However, as a result of the short decorrelation timescale in the prior ensemble, the spread of the ensemble forecast increases quickly with time. Throughout the paper, a special emphasis is given to discussing the statistical reliability of the solution. Two different methods have been applied to perform this 4D statistical analysis and forecast: the analysis step of the ensemble transform Kalman filter (with domain localization) and a Monte Carlo Markov chain (MCMC) sampler (with covariance localization), both enhanced by the application of anamorphosis to the original variables. Despite being very different, the two algorithms produce very similar results, thus providing support to each other's estimates. As shown in the paper, the decoupling of the statistical analysis from the dynamical model allows us to restrict the analysis to a few selected variables and, at the same time, to produce estimates of additional ecological indicators (in our example: phenology, trophic efficiency, downward flux of particulate organic matter). This approach can easily be appended to existing operational systems to focus on dedicated users' requirements, at a small additional cost, as long as a reliable prior ensemble simulation is available. It can also serve as a baseline to compare with the dynamical ensemble forecast and as a possible substitute whenever useful.

What is the basis of ensemble subset selection?

The visual system can rapidly calculate the ensemble statistics of a set of objects; for example, people can easily estimate an average size of apples on a tree. To accomplish this, it is not always useful to summarize all the visual information. If there are various types of objects, the visual system should select a relevant subset: only apples, not leaves and branches. Here, we ask what kind of visual information makes a "good" ensemble that can be selectively attended to provide an accurate summary estimate. We tested three candidate representations: basic features, preattentive object files, and full-fledged bound objects. In four experiments, we presented a target and several distractors' sets of differently colored objects. We found that conditions where a target ensemble had at least one unique color (basic feature) provided ensemble averaging performance comparable to the baseline displays without distractors. When the target subset was defined as a conjunction of two colors or color-shape partly shared with distractors (so that they could be differentiated only as preattentive object files), subset averaging was also possible but less accurate than in the baseline and feature conditions. Finally, performance was very poor when the target subset was defined by an exact feature relationship, such as in the spatial conjunction of two colors (spatially bound object). Overall, these results suggest that distinguishable features and, to a lesser degree, preattentive object files can serve as the representational basis of ensemble selection, while bound objects cannot.

Technical note: Accurate, reliable, and high-resolution air quality predictions by improving the Copernicus Atmosphere Monitoring Service using a novel statistical post-processing method

Abstract. Starting from the regional air quality forecasts produced by the Copernicus Atmosphere Monitoring Service (CAMS), we propose a novel post-processing approach to improve and downscale results on a finer scale. Our approach is based on the combination of ensemble model output statistics (EMOS) with a spatio-temporal interpolation process performed through the stochastic partial differential equation–integrated nested laplace approximation (SPDE-INLA). Our interpolation approach includes several spatial and spatio-temporal predictors, including meteorological variables. A use case is provided that scales down the CAMS forecasts on the Italian peninsula. The calibration is focused on the concentrations of several air quality pollutants (PM10, PM2.5, NO2, and O3) at a daily resolution from a set of 750 monitoring sites, distributed throughout the Italian country. Our results show the key role that conditioning variables play in improving the forecast capabilities of ensemble predictions, thus allowing for a net improvement in the calibration with respect to ordinary EMOS strategies. From a deterministic point of view, the performance of the predictive model shows a significant improvement in the performance of the raw ensemble forecast, with an almost-zero bias, significantly reduced root mean square errors, and correlations that are almost always higher than 0.9 for each pollutant; moreover, the post-processing approach is able to significantly improve the prediction of exceedances, even for very low thresholds, such as those recently recommended by the World Health Organisation. This is particularly significant if a forecasting approach is used to predict air quality conditions and plan adequate human health protection measures, even for low alert thresholds. From a probabilistic point of view, the quality of the forecast was verified in terms of reliability and credible intervals. After post-processing, the predictive probability density functions were sharp and much better calibrated than the raw ensemble forecast. Finally, we present some additional results based on a set of gridded (4 km × 4 km) maps covering the entire Italian country for the detection of areas where pollution peaks occur (exceedances of the current and/or proposed regulatory thresholds).

The functional role of spatial anisotropies in ensemble perception.

The human brain can rapidly represent sets of similar stimuli by their ensemble summary statistics, like the average orientation or size. Classic models assume that ensemble statistics are computed by integrating all elements with equal weight. Challenging this view, here, we show that ensemble statistics are estimated by combining parafoveal and foveal statistics in proportion to their reliability. In a series of experiments, observers reproduced the average orientation of an ensemble of stimuli under varying levels of visual uncertainty. Ensemble statistics were affected by multiple spatial biases, in particular, a strong and persistent bias towards the center of the visual field. This bias, evident in the majority of subjects and in all experiments, scaled with uncertainty: the higher the uncertainty in the ensemble statistics, the larger the bias towards the element shown at the fovea. Our findings indicate that ensemble perception cannot be explained by simple uniform pooling. The visual system weights information anisotropically from both the parafovea and the fovea, taking the intrinsic spatial anisotropies of vision into account to compensate for visual uncertainty.

A two‐step machine‐learning approach to statistical post‐processing of weather forecasts for power generation

By the end of 2021, the renewable energy share of the global electricity capacity reached 38.3% and the new installations were dominated by wind and solar energy, showing global increases of 12.7% and 18.5% respectively. However, both wind and photovoltaic energy sources are highly volatile, making planning difficult for grid operators; thuss, accurate forecasts of the corresponding weather variables are essential for reliable electricity predictions. The most advanced approach in weather prediction is the ensemble method, which opens the door for probabilistic forecasting. However, ensemble forecasts are often underdispersive and subject to systematic bias. Hence, they require some form of statistical post‐processing, where parametric models provide full predictive distributions of the weather variables at hand. We propose a general two‐step machine‐learning‐based approach to calibrating ensemble weather forecasts, where, in the first step, improved point forecasts are generated, which then together with various ensemble statistics serve as input features of the neural network estimating the parameters of the predictive distribution. In two case studies based on 100 m wind speed and global horizontal irradiance forecasts of the operational ensemble prediction system of the Hungarian Meteorological Service, the predictive performance of this novel method is compared with the forecast skill of the raw ensemble and the state‐of‐the‐art parametric approaches. Both case studies confirm that, at least up to 48 hr, statistical post‐processing substantially improves the predictive performance of the raw ensemble for all forecast horizons considered. The variants of the proposed two‐step method investigated outperform in skill their competitors, and the suggested new approach is well applicable for different weather quantities and for a fair range of predictive distributions.